Question

Describe the solid using spherical and cylindrical coordinates

Answer 1

Start with a sphere. In spherical coordinates, a sphere of radius \(r\) is described by the set
\[S_\text{sph}=\left\{(\rho,\theta,\varphi)~:~0\le\rho\le r,~0\le\theta\le2\pi,~0\le\varphi\le\pi\right\}\]
You're given that the radius is \(r=3\).

The azimuthal angle \(\theta\) (the angle any point within the region makes with the reference "horizon", which in this case would be the angle made by the point's shadow in the \(x\)-\(y\) plane and the \(x\) axis) determines how far around the \(z\) axis points in the solid are distributed. In this case, the solid occupies space that amounts to a full revolution about the \(z\) axis, so \(0\le\theta\le2\pi\).

The polar angle \(\varphi\) can be determined with basic trig:

This means \(\tan\varphi=\dfrac{1}{\sqrt8}\), and so \(0\le\varphi\le\arctan\left(\dfrac{1}{\sqrt8}\right)\).